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It is well known that there exists a Quillen equivalence,

$$\mathfrak{C}: Set_{\Delta} \rightleftarrows Cat_{\Delta}: N, \enspace \mathfrak{C} \dashv N$$

between Joyal's model structure on simplicial sets and Bergner's model structure on simplicially enriched categories. I will call the left adjoint the rigidification functor.

Since every object is cofibrant in Joyal's model structure this Quillen equivalence can be used to show when a map of simplicial sets is a categorical equivalence by checking this after passing to simplicial categories.

I was wondering if there exists an analogous result for marked simplicial sets.

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  • $\begingroup$ If you are happy with only detecting them between fibrant marked simplicial sets then you can simply forget the marking and apply the same argument :) $\endgroup$ – Edoardo Lanari Aug 22 '19 at 14:08
  • $\begingroup$ It should be true that a map of marked simplicial sets is a weak equivalence if and only if the induced functor on localizations of simplicial categories (where marked arrows are localized) is an equivalence. $\endgroup$ – Valery Isaev Aug 22 '19 at 17:35
  • $\begingroup$ @ValeryIsaev Since this is a model-dependent question, there's more to say, right? Do you have a particular model for the localization in mind? If you just take DK localization, then you end up sending $\Delta[1]^\sharp$ to the constant simplicial category at the walking isomorphism, which is not Bergner-cofibrant, so you don't get a left Quillen functor. $\endgroup$ – Tim Campion Sep 23 '19 at 18:47
  • $\begingroup$ @TimCampion Localization is defined uniquely up to equivalence, so it does not matter which one we use. The functor has to be homotopical, but it does not have to be a left Quillen functor. Actually, I can prove my claim, so I'll post it as an answer. $\endgroup$ – Valery Isaev Sep 24 '19 at 12:17
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Let $f : X \to Y$ be a map of marked simplicial sets. Then $f$ is a weak equivalence if and only if the induced functor on localizations of simplicial categories (where marked arrows are localized) is an equivalence.

Indeed, $f$ is a weak equivalence if and only if $R(f)$ is a weak equivalence, where $R$ is a fibrant replacement functor in the model category on marked simplicial sets. Since the forgetful functor from marked simplicial sets to simplicial sets with the Joyal model structure is the right part of a Quillen equivalence, $R(f)$ is a weak equivalence if and only if $R(f)$ induces a weak equivalence $U(R(f))$ of underlying quasicategories. Since $\mathfrak{C}$ is the left part of a Quillen equivalence, this map is a weak equivalence if and only if $\mathfrak{C}(U(R(F)))$ is a weak equivalence of simplicial categories. Thus, we just need to show that $\mathfrak{C}(U(R(X)))$ is the localization of $\mathfrak{C}(U(X))$ at the set of marked maps.

The underlying quasicategory of $R(X)$ can be defined as the localization of $X$ as described in section 1.1.3 of Dwyer-Kan localization revisited, Vladimir Hinich. The claim follows from the fact that functor $\mathfrak{C}$ maps the localization of a simplial set to DK localozation of the corresponding simplicial category, which is proved in the same paper right before Proposition 1.2.1.

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  • $\begingroup$ Ah, I see! Hinich observes that the localization of $C$ at $W$ is $C \cup_W W[W^{-1}]$, and then you don't need to take a Joyal-fibrant replacement of this before applying $\mathfrak C$. To be really concrete, you could model $W[W^{-1}]$ by $Ex^\infty(W)$ -- or even just glue in a left and right inverse to each 1-cell of $W$. $\endgroup$ – Tim Campion Sep 24 '19 at 13:15

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